Local rigidity of group actions: past, present, future

Summary

Abstract. This survey aims to cover the motivation for and history of the study of local rigidity of group actions. There is a particularly detailed discussion of recent results, including outlines of some proofs. The article ends with a large number of conjectures and open questions and aims to point to interesting directions for future research.

Prologue

Let Г be a finitely generated group, D a topological group, and π : Г → D a homomorphism. We wish to study the space of deformations or perturbations of π. Certain trivial perturbations are always possible as soon as D is not discrete, namely we can take d π d^{– 1} where d is a small element of D. This motivates the following definition:

Definition 1.1. Given a homomorphism π : Г → D, we say π is locally rigid if any other homomorphism π′ which is close to π is conjugate to π by a small element of D.

We topologize Hom(Г, D) with the compact open topology which means that two homomorphisms are close if and only if they are close on a generating set for Г. If D is path connected, then we can define deformation rigidity instead, meaning that any continuous path of representations π_t starting at π is conjugate to the trivial path π_t = π by a continuous path d_t in D with d₀ being the identity in D. If D is an algebraic group over ℝ or ℂ, it is possible to prove that deformation rigidity and local rigidity are equivalent since Hom.(Г, D) is an algebraic variety and the action of D by conjugation is algebraic; see [Mu], for example.